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Let $A = M_n(\mathbb C)$. Then it is possible to endow this $\ast$-algebra with several different norms (see here):

$$ \|a\|_1 = \max_j \sum_i |a_{ij}|$$

$$ \|a\|_\infty = \max_i \sum_j |a_{ij}|$$

and the operator norm:

$$ \|a\| = \sup_{\|v\|=1}\|Av\|$$

It is pointed out in the Wikipedia article that the operator norm is different from the $1$- and the $\infty$-norm.

Assuming that all three norms above make $A$ a Banach algebra, I was wondering which of them makes it also a $C^\ast$-algebra? (there can be only one)

On a related note, would $\displaystyle \|a\|_p = \left ( \max_j \sum_i |a_{ij}|^p \right)^{1 \over p}$ and $\displaystyle \|a\|_p = \left ( \max_i \sum_j |a_{ij}|^p \right)^{1 \over p}$ also define matrix norms?

1 Answers1

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It is the operator norm. It is the only one that satisfies $\|a^*a\|=\|a\|^2$ for all $a$. Actually, you obtain the operator norm precisely when you represent $M_n(\mathbb C)$ as the algebra of linear operators on $\mathbb C^n$.

Martin Argerami
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  • Again, thank you for your (always) helpful answers. Could you, if you don't mind, add something about my $p$-norms at the bottom of my question? Do they define valid matrix norms (to make it a Banach algebra, maybe)? –  Dec 01 '14 at 05:37